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Mirrors > Home > ILE Home > Th. List > ax-mulass | GIF version |
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7886. Proofs should normally use mulass 7956 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7823 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 2158 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2158 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 2158 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 979 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 7830 | . . . . 5 class · | |
10 | 1, 4, 9 | co 5888 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 5888 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 5888 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 5888 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1363 | . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
15 | 8, 14 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff set class |
This axiom is referenced by: mulass 7956 |
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